1. Field of the Invention
The present invention generally relates to method, system and software product used in structural design using finite element analysis, more particularly to distinguish effects due to bifurcation from effects due to design variable changes in a structural design study.
2. Description of the Related Art
Finite element analysis (FEA) is a computerized method widely used in industry to model and solve engineering problems relating to complex systems. FEA derives its name from the manner in which the geometry of the object under consideration is specified. With the advent of the modern digital computer, FEA has been implemented as FEA software. Basically, the FEA software is provided with a model of the geometric description and the associated material properties at each point within the model. In this model, the geometry of the system under analysis is represented by solids, shells and beams of various sizes, which are called elements. The vertices of the elements are referred to as nodes. The model is comprised of a finite number of elements, which are assigned a material name to associate with material properties. The model thus represents the physical space occupied by the object under analysis along with its immediate surroundings. The FEA software then refers to a table in which the properties (e.g., stress-strain constitutive equation, Young's modulus, Poisson's ratio, thermo-conductivity) of each material type are tabulated. Additionally, the conditions at the boundary of the object (i.e., loadings, physical constraints, etc.) are specified. In this fashion a model of the object and its environment is created.
FEA had it beginnings as a method for structural analysis, but today is routinely used in the design of motors, generators, magnetic resonance imaging systems, aircraft engine ignition systems, circuit breakers and transformers, to name but a few; its techniques are used to analyze stress, temperature, molecular structure, electromagnetic fields, car crash, metal stamping, physical forces, etc. in all sorts of physical systems. It has become a standard part of the design cycle for numerous products which are not easily analyzed by other methods.
FEA is becoming increasingly popular with automobile manufacturers for optimizing both the aerodynamic performance and structural integrity of vehicles. Similarly, aircraft manufacturers rely upon FEA to predict airplane performance long before the first prototype is built. Rational design of semiconductor electronic devices is possible with Finite Element Analysis of the electrodynamics, diffusion, and thermodynamics involved in this situation. FEA is utilized to characterize ocean currents and distribution of contaminants. FEA is being applied increasingly to analysis of the production and performance of such consumer goods as ovens, blenders, lighting facilities and many plastic products. In fact, FEA has been employed in as many diverse fields as can be brought to mind, including plastics mold design, modeling of nuclear reactors, analysis of the spot welding process, microwave antenna design, simulating of car crash and biomedical applications such as the design of prosthetic limbs. In short, FEA is utilized to expedite design, maximize productivity and efficiency, and optimize product performance in virtually every stratum of light and heavy industry. This often occurs long before the first prototype is ever developed.
Generally, FEA begins by generating a finite element model of a system. In this model, a subject structure is reduced into a number of node points which are connected together to form finite elements. Governing equations of motion are written in a discrete form, where the motions of each node point are the unknown part of the solution. A simulated load or other influence is applied to the system and the resulting effect is analyzed using well known mathematical methods.
To design a structure, engineers study the effects of modifying certain design variables (e.g., the thickness of a plate, the cross-section area of a beam, or angle of loading direction). Design methods can use metamodels to predict the structural responses. The metamodel is constructed using the FEA solutions obtained for a selected design cases via a number of mathematical techniques, such as least squares fitting, Taylor series expansion, neural nets and Kringing approximations. In particular, the metamodel created with least squares fitting is called a response surface. Engineers can select an improved design using the metamodels. Today, not only has metamodels been applied to simple structures, it has also been used for very complicated, highly non-linear, impact analysis (e.g., car crash simulation). In theory, the design study can be performed without much difficulty. However, when there is a bifurcation in the FEA solution, it creates a huge problem for the design study due to multiple valid solutions for a given set of design variables. The most common bifurcation in structural design is an instability called buckling.
Buckling is an instability occurring when the loading of the structure exceeds a certain critical value. When the loading reaches the critical value, the structure will become unstable or buckle. A feature of buckling is that there exists more than one mode. The structure can buckle in different directions or different modes. Sometimes the structure experiences local buckling in different forms of wrinkles or corrugations. In real world, the structure usually buckles in a direction determined by initial imperfections of the structure instead of an arbitrary direction that theoretical solution predicts. The numerical simulation of the structural buckling in FEA software at times reflects the existence of more than one valid buckling mode and different initial imperfections—depending on the digital computer and the operating system, a tiny difference in the floating point number may result in the buckling of structure in different directions. Different bifurcations may also be triggered by, but not due to, design variable changes. In general a design variable change will cause a change in the results and potentially cause the computational algorithms to follow a different bifurcation. For very complicated, highly non-linear problems such as car crash simulation, the chances of having a different bifurcation are significant. It is therefore difficult to be sure whether a change in results is due to a change in the design variable values or different bifurcation.
There is no logical method to distinguish which FEA result is a likely candidate associated with bifurcation. It is therefore desirable to have a new method to distinguish these effects more efficiently and effectively.